The higher-weight Elliptic Stark conjecture

Let $(f,g,h)$ have weights $(k,\ell,m)$ with $k\geq\ell+m$ and $k,\ell,m\geq2$, and let $M(f\otimes g\otimes h)$ be the associated triple-product motive. Put $c=(k+\ell+m-2)/2$ and let $\operatorname{Reg}(f,g,h)$ be the regulator defined from a basis of $\operatorname{CH}^c(M(f\otimes g\otimes h))_{0,L}$. Let ${\bf f},{\bf g},{\bf h}$ be the corresponding Hida families and set $r=\dim_L\operatorname{CH}^c(M(f\otimes g\otimes h))_{0,L}$. Higher-weight Elliptic Stark conjecture. If $r>2$, then

$$\mathcal{L}^g_p(\breve{\bf f},\breve{\bf g},\breve{\bf h})(k,\ell,m)=0$$

for every choice of test vectors. If $\operatorname{ord}_{s=c}L(f\otimes g\otimes h,s)=2$, then there exist a finite extension $L_0$ of $L$, suitable test vectors, and Hida families specializing to them such that

$$\mathcal{L}^g_p(\breve{\bf f},\breve{\bf g},\breve{\bf h})(k,\ell,m)=\operatorname{Reg}(f,g,h)\pmod{L_0^\times}.$$

This is proposed as the higher-weight analogue of the Elliptic Stark conjecture, relating triple-product $p$-adic $L$-values to motivic regulators; it remains conjectural in the source.