This question is about constructive mathematics, without any form of Choice except Unique Choice, such as in the internal logic of a topos with natural numbers object, or in IZF. The “reals” (and the symbol $\mathbb{R}$) refer to the Dedekind real numbers.
Recollections: The following definitions and facts should be standard but are repeated here lest there be any ambiguity as to their meaning. Experts should skip directly to the question below.
$\textbf{LLPO}_{\mathbb{R}}$ (or the “Analytic Lesser Limited Principle of Omniscience”) is the statement that every real number is $\leq 0$ or $\geq 0$:$$\forall x\in\mathbb{R}.(x\leq 0 \lor x\geq 0)$$
$\textbf{WLLPO}_{\mathbb{R}}$ (or the “Analytic Weak Lesser Limited Principle of Omniscience”) is the statement that every real number that is not $=0$ is $\leq 0$ or $\geq 0$:$$\forall x\in\mathbb{R}.(\neg(x=0) \; \Rightarrow \; x\leq 0 \lor x\geq 0)$$
$\textbf{LLPO}_{\mathrm{seq}}$ (or the “Sequential Lesser Limited Principle of Omniscience”) is the statement that in a binary sequence with at most one $1$ either the even terms are all zero or the odd terms are all zero:$$\forall u\in\mathbb{N}_\infty.((\forall k.u_{2k}=0) \lor (\forall k.u_{2k+1}=0))$$Here $\mathbb{N}_\infty := \{u\in \{0,1\}^{\mathbb{N}} : \forall m,n.(u_m=u_n=1 \Rightarrow m=n)\}$.
$\textbf{WLLPO}_{\mathrm{seq}}$ (or the “Sequential Weak Lesser Limited Principle of Omniscience”) is the statement that in a binary sequence with at most one $1$ that is not identically zero either the even terms are all zero or the odd terms are all zero:$$\forall u\in\mathbb{N}_\infty.(\neg(\forall n(u_n=0)) \; \Rightarrow \; (\forall k.u_{2k}=0) \lor (\forall k.u_{2k+1}=0))$$(This one isn't really relevant to the question.)
(Both the “$\mathbf{WLLPO}$” variants are often called by the term “disjunctive Markov principle”, $\mathbf{MP}^\lor$ instead.)
We have the following implications:$$\begin{array}{ccc}\textbf{LLPO}_{\mathbb{R}} & \Longrightarrow & \textbf{WLLPO}_{\mathbb{R}}\\\Downarrow&&\Downarrow\\\textbf{LLPO}_{\mathrm{seq}} & \Longrightarrow & \textbf{WLLPO}_{\mathrm{seq}}\end{array}\tag{*}$$The horizontal implications are trivial. The vertical ones are proven by considering the real number $\sum_{n=0}^{+\infty} (-\frac{1}{2})^n\,u_n$ for $u \in \mathbb{N}_\infty$.
In the presence of a modicum of Choice, the vertical implications are equivalences. The axiom of Choice for sequences of inhabited subsets of $\{0,1\}$ suffices for this. (The idea is that, for any $x\in\mathbb{R}$, we have either $(-1)^n\,x < 2^{-n}$ or $(-1)^n\, x > 0$, so we can use this form of Choice to construct a binary sequence $(p_n)$ such that $p_n=0$ implies the former and $p_n=1$ implies the latter; now let $u_n$ be $1$ when $p_n=1$ and $p_k=0$ for all $k<n$, and $0$ otherwise. If $(p_{2k})$, resp. $(p_{2k+1})$ is identically $0$ then $x\leq 0$, resp. $x\geq 0$.)
None of the implications in $(*)$ can be reversed. Indeed, the topos of sheaves over $\mathbb{R}$ satisfies $\textbf{LLPO}_{\mathrm{seq}}$ (in fact, it even satisfies $\textbf{LPO}_{\mathrm{seq}}$, namely that every $u\in\mathbb{N}_\infty$ is either identically $0$ or has a term equal to $1$) but does not satisfy $\textbf{WLLPO}_{\mathbb{R}}$, which shows that the vertical implications cannot be reversed. And the effective topos satisfies all forms of $\mathbf{WLLPO}$ but no form of $\mathbf{LLPO}$, which shows that the horizontal implications cannot be reversed (the effective topos satisfies Countable Choice, so the previous point implies that the various forms of $\mathbf{LLPO}$, resp. of $\mathbf{WLLPO}$, are equivalent in it).
All of this raises the question:
✱ QUESTION: does $\textbf{WLLPO}_{\mathbb{R}}$ together with $\textbf{LLPO}_{\mathrm{seq}}$ imply $\textbf{LLPO}_{\mathbb{R}}$?
I don't know how to produce countermodels for this: realizability models will tend to satisfy Choice, as well as Markov's principle, which is stronger than $\textbf{WLLPO}$ anyway, and getting a topological model to satisfy $\textbf{WLLPO}_{\mathbb{R}}$ but not $\textbf{LLPO}_{\mathbb{R}}$ (regardless of sequential variants) seems difficult. But I also have no idea how a positive result could be proved.
Note that this other question (about the relation between $\forall x\in\mathbb{R}.(\forall r\in\mathbb{Q}.\neg(x=r) \, \Rightarrow \, x\leq 0 \lor x\geq 0)$ to $\textbf{WLLPO}$) seems to be even more delicate, so I'm putting the present one as a hopefully less inaccessible goal.
