Maturana–Rashmi bandwidth-optimality conjecture for split-regime MDS convertible codes

Let $n^{I},k^{I};n^{F},k^{F}$ be the parameters of a stable linear MDS convertible code, with $k^{I}={bb}^{F}k^{F}$, where ${bb}^{F}\geq 2$. Write $r^{I}=n^{I}-k^{I}$ and $r^{F}=n^{F}-k^{F}$. Under the Uniform Cost Assumption, the coordinator downloads the same number of symbols from each information node and the same number of symbols from each parity node. Let $\alpha$ be the node storage size and let $\gamma_{\mathrm{R}}$ denote the read bandwidth cost.

Maturana–Rashmi's conjecture. Under the Uniform Cost Assumption, every stable linear MDS $(n^{I},k^{I}={\lambda}^{F}k^{F};n^{F},k^{F})$ convertible code with $r^{F}<r^{I}$ and $r^{F}<k^{F}$ satisfies

$$\gamma_{\mathrm{R}}\geq {\lambda}^{F}r^{F}\alpha\frac{({\lambda}^{F}-1)k^{F}+r^{I}}{({\lambda}^{F}-1)r^{F}+r^{I}}.$$

This conjectured lower bound would establish that the constructions of Maturana and Rashmi are bandwidth-optimal under the Uniform Cost Assumption in the regime $r^{F}<k^{F}$ and $r^{F}<r^{I}$; the source presents this as an open problem.