The Functional Batch Conjecture for binary simplex codes

Let the binary $[k,2^k-1]$ simplex code be the linear code of dimension $k$ and length $2^k-1$ over $\mathbb F_2$ whose generator matrix has one representative of each nonzero vector of $\mathbb F_2^k$ as a column. A code is a $t$-functional batch code if it supports simultaneous retrieval of any $t$ requested linear functions of the stored data.

Functional Batch Conjecture. The binary $[k,2^k-1]$ simplex code is a $2^{k-1}$-functional batch code.

This is presented as a central open problem in functional batch coding. The supplied text gives no resolution status or further general result for this assertion.