Potentially diagonalisable lifts with controlled Hodge--Tate weights

Motivated by the weight part of Serre’s conjecture we consider the following question. Let $K\mathrm{/}{\mathbb{Q}}_p$ be a finite extension and suppose $\overline{\rho} \colon G_K \rightarrow \operatorname{GL}_n(\overline{$\mathbb{F}$}_p)$ admits a crystalline lift with Hodge–Tate weights contained in the range $[0,p]$. Does $\bar{\rho }$ admits a potentially diagonalisable crystalline lift of the same Hodge–Tate weights? We answer this question in the affirmative when $K={\mathbb{Q}}_p$ and $n\le 5$, and $\bar{\rho }$ satisfies a mild `cyclotomic-free’ condition. We also prove partial results when $K\mathrm{/}{\mathbb{Q}}_p$ is unramified and $n$ is arbitrary.