From Vitali to Heine-Borel in reverse mathematics

The Vitali and Heine-Borel covering theorems are house-hold names of analysis, and rightly well-studied in reverse mathematics. As shown in Simpson's excellent monograph [1], for countable coverings of the unit interval, the Heine-Borel theorem is equivalent to WKL (weak Koenig's lemma), while the Vitali covering theorem is equivalent to WWKL (weak weak Koenig's lemma). The theorem numbers in [1] are IV.1.2 and X.1.13.

My question is then as follows:

Is there a natural statement X such that [WWKL +X ] $$\leftrightarrow$$ WKL, say over RCA$$_0$$?

Here, $$X$$ should be weaker than WKL, obviously. Results in related frameworks (computability theory, Weihrauch reducibility, constructive math, ...) are also welcome.

PS: I am asking this question because in the case of uncountable coverings, such an X does exist.

[1] Stephen G. Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, Cambridge University Press, 2009.

First, we are looking at principles below $$\mathrm{WKL}$$ which are incomparable with $$\mathrm{WWKL}$$. To my knowledge, the only such principles which were studied in the literature are variants of convex choice $$\mathrm{XC}_{[0,1]^n}$$. All these variants are themselves below the principle $$\mathrm{XC}_{[0,1]}^\diamond$$, which lets us invoke choice for subsets of the unit interval finitely many times (with subsequent queries depending on previous answers). Note that $$\mathrm{XC}_{[0,1]}$$ is equivalent to the intermediate value theorem.
However, $$(\mathrm{XC}_{[0,1]} \sqcup \mathrm{WWKL})^\diamond <_{\mathrm{W}} \mathrm{WKL}$$ (the left hand side is "make finitely many calls to both $$\mathrm{XC}_{[0,1]}$$ and to $$\mathrm{WWKL}$$). The reason for this is that on a computable input, $$\mathrm{XC}_{[0,1]}$$ can always return something computable, and $$\mathrm{WWKL}$$ can always return some ML-random. Thus, the left-hand side can always return something which is computable from a ML-random, but this doesn't hold for $$\mathrm{WKL}$$, as $$\mathrm{PA}$$-degrees are not computable from ML-randoms.
Another piece of evidence is that there is multivalued function $$g$$ at all such that $$\mathrm{WKL} \leq_{\mathrm{W}} \mathrm{WWKL} \circ g$$. On the right, we have that every input has a positive measure of potential solutions, and $$\mathrm{WKL}$$ has instances where the Turing-upper cone of solutions is measure $$0$$.