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.