Puncturing-inclusion conjecture for MRD codes

Let $n,d\in\mathbb N$ satisfy $1<d<n$, and let $\mathbb F$ be a finite field. An MRD code is a maximum rank-distance code. Puncturing-inclusion conjecture. There exists an $[n\times(n-1),n(n-d),d]_{\mathbb F}$ MRD code $\mathcal C$ such that its puncturing on one row is contained in an $[(n-1)\times(n-1),(n-1)(n-d+1),d-1]_{\mathbb F}$ MRD code. This open problem is equivalent to a smaller instance of the Etzion–Silberstein conjecture.