The Duffin–Schaeffer-type conjecture for multiple zeta-star values

Let $f\colon \mathbb{N}\rightarrow [1,+\infty)$, and let $B_f$ be the set of real numbers $\alpha\in [1,+\infty)$ such that

$$\left|\alpha-\zeta^\star(k_1,\ldots,k_r)\right|<\frac{1}{2^{k_1+\cdots+k_r+f(r)}}$$

for infinitely many $r,k_1,\ldots,k_r$ with $r\rightarrow +\infty$. The Duffin–Schaeffer-type conjecture. If

$$\sum_{r\geq 1}\frac{1}{2^{f(r)}}$$

is divergent, then

$$m\bigl((1,+\infty)-B_f\bigr)=0.$$

This conjecture is the divergent-series counterpart to the preceding zero-one-law result, which proves measure zero when the series converges. It asks whether almost every real number in $(1,+\infty)$ admits infinitely many approximations by multiple zeta-star values at the stated scale.