For each $n=0,1,2,\ldots$, the central trinomial coefficient $T_n$ is defined as the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. It is easy to see that $T_n=\sum_{k=0}^{\lfloor n/2\rfloor}\binom{n}{2k}\binom{2k}k$.

On December 7, 2019, I conjectured that
$$\sum_{k=1}^\infty\frac{(105k-44)T_{k-1}}{k^2\binom{2k}k^23^{k-1}}=\frac{5\pi}{\sqrt3}+6\log3\tag{1}$$
and
$$\sum_{k=2}^\infty\frac{(5k-2)T_{k-1}}{k^2\binom{2k}k^2(k-1)3^{k-1}}=\frac{21-2\sqrt3\,\pi-9\log3}{12}.\tag{2}$$
As the two series converge very fast, it is easy to check (1) and (2) numerically. The two identities and related congruences appear in Section 10 of my recent preprint *New series for powers of $\pi$ and related congruences*. I'm unable to find proofs of $(1)$ and $(2)$. So, here I ask the following question.

**Question.** How to prove the conjectural identities $(1)$ and $(2)$?

Your comments are welcome!