Simplex-code conjecture for functional batch codes

Let $r$ be a positive integer, and let the $[2^r-1,r]$ simplex code be the binary simplex code of length $2^r-1$ and dimension $r$. Let $FB(r,k)$ denote the minimum number of servers in a functional $k$-batch code with $r$ information symbols. Simplex-code conjecture. The $[2^r-1,r]$ simplex code is a functional $2^{r-1}$-batch code, and consequently

$$FB(r,2^{r-1})=2^r-1.$$

This is presented as a conjecture motivated by constructions of parallel RIO and functional batch codes; the stated equality is verified in the examples discussed for $r=3,4,5$, while the general assertion remains open.