Weight-derivative factorization conjecture for an Eisenstein family

Let $r$ be an integer, let $\tau$ and $\psi$ be Dirichlet characters, let ${\mathbf{f}}$ denote the relevant Eisenstein family, and let ${\mathbf j}$ denote the cyclotomic-variable notation. Assume that either $r$ is even and $\tau$ is odd, or that $r$ is odd and $\tau$ is also odd. Let $L_p'({\mathbf f})(r)$ be the weight derivative of the $p$-adic $L$-function specialized at weight $r$, let $\lambda$ be the previously defined Iwasawa function, and let $\log^{[r+1]}$ be the indicated logarithmic factor. Weight-derivative factorization. If $\tau$ is odd—either under the stated even-$r$ assumption or when $r$ is odd and $\tau$ is also odd—then

$$L_p'({\mathbf f})(r)=\lambda\cdot\log^{[r+1]}L_p(\psi,{\mathbf j}+1)\cdot L_p(\tau,{\mathbf j}-r).$$

The formula is motivated by the vanishing of $L_p(f_{\beta})$ and is intended as an analogue of Bellaïche--Dasgupta's factorization into Kubota--Leopoldt $p$-adic $L$-functions. Its proof is stated to be beyond the techniques used in the paper, although the authors discuss theoretical evidence for it.