Finite-file PIR rate upper-bound conjecture for coded colluding servers

Let ${\mathcal{C}}$ be an $[n,k,d]$ code with generator matrix ${\mathbf{G}}$, storing $m$ files through the distributed storage system

$${\mathbf{Y}}={\mathbf{X}}\cdot{\mathbf{G}}.$$

Fix $1\leq t\leq n-k$, and let $R_m$ denote the rate of a PIR scheme for ${\mathbf{Y}}$ protecting against any $t$ colluding servers.

Finite-file PIR rate conjecture. Any such PIR scheme has rate at most

$$R_m\leq\frac{1-\frac{k+t-1}{n}}{1-\left(\frac{k+t-1}{n}\right)^m} \mathrel{\xrightarrow[m\rightarrow\infty]{}}1-\frac{k+t-1}{n}.$$

For finitely many files, this conjectured upper bound is motivated by the capacity expressions known when either $k=1$ or $t=1$, and its limiting value agrees with the corresponding asymptotic capacity. Its resolution is not established in the supplied text.