Crystalline liftability conjecture for irregular weights

Let $K$ be the finite unramified extension of $\mathbb Q_p$ used in the paper, let $G_K$ be its absolute Galois group, and let $\Sigma$ be the corresponding set of embeddings. Let $(k,0)$ be an irregular weight with $1\leq k_{\tau}\leq p$ for every $\tau\in\Sigma$, and assume $k\ne1$. Let $(k',l')$ and $({k}^{\Theta,\text{alt}},{l}^{\Theta,\text{alt}})$ be the weights defined in the paper. Crystalline liftability conjecture. A Galois representation $\rho:G_K\to\operatorname{GL}_2(\overline{\mathbb F}_p)$ has a crystalline lift of weight $(k,0)$ if and only if it has a crystalline lift of weight $(k',l')$ and a crystalline lift of weight $({k}^{\Theta,\text{alt}},{l}^{\Theta,\text{alt}})$. The conjecture is motivated by translating regular-weight crystalline liftability results to irregular weights and remains open.