  We prove a new (conditional) result towards the subconvexity problem for
certain automorphic $L$-functions for $\mathrm{GL}_2 \times \mathrm{GL}_3$.
This follows from the computation of new $\mathrm{SL}_2$-period integrals
associated with newforms $f$ and $g$ of even weight and odd squarefree level.
The same computations also lead to a central value formula for degree $6$
complex $L$-series of the form $L(f \otimes \mathrm{Ad}(g), s)$, extending
previous work in arXiv:1804.02352.
