Nonvanishing conjecture for critical $p$-adic $L$-functions

Throughout, let $f_\alpha$ be a $\theta$-critical normalized eigenform, and let $L_{\mathrm{S},\alpha}^{[0]}(f,\xi)$ and $L_{\mathrm{K},\alpha}(f,\xi)$ denote the corresponding critical $p$-adic $L$-functions. Nonvanishing conjecture. The functions $L_{\mathrm{S},\alpha}^{[0]}(f,\xi)$ and $L_{\mathrm{K},\alpha}(f,\xi)$ are nonzero.

This is an open problem in the theory of critical $p$-adic $L$-functions, arising from the comparison between the constructions of Bellaïche and the authors in an infinitesimal neighborhood of a $\theta$-critical point.