Weight-derivative factorization for the beta $p$-adic $L$-function

Let $r$ be an integer and let $\tau$ and $\psi$ be Dirichlet characters satisfying $(-1)^s=-\psi(-1)$. Let $L_p'(f_{\beta})(s)$ denote the weight derivative of the beta $p$-adic $L$-function, let $\lambda$ be the previously defined function over the Iwasawa algebra, let $N_2$ be the associated integer, and let $\log^{[r+1]}$ denote the indicated logarithmic factor. Weight-derivative factorization. If $(-1)^s=-\psi(-1)$, then, up to multiplication by some explicit constant,

$$L_p'(f_{\beta})(s)=\lambda N_2^{-s}\log^{[r+1]}\cdot L_p(\psi,s)\cdot L_p(\tau,s-r-1).$$

This is proposed as a natural analogue of Bellaïche's secondary $p$-adic $L$-functions and as an extension of work of Bellaïche--Dasgupta; it describes the factorization expected when the original $p$-adic $L$-function vanishes and its weight derivative is studied. The available methods do not prove the formula.