Sun's 1/pi series conjecture for S_n^{(2)}(1/11) and S_n^{(2)}(1/18)

Let

$$S_n^{(2)}(x)=\sum_{i=0}^n\sum_{j=0}^n\binom ni^2\binom nj^2\binom{i+j}i x^{i+j}.$$

The two displayed series are conjectured to converge to the stated multiples of $1/\pi$. Sun's 1/pi series conjecture.

$$\sum_{k=0}^\infty(7k+1)\frac{S_k^{(2)}(1/11)}{9^k}=\frac{5445}{104\sqrt{39}\,\pi}$$

and

$$\sum_{k=0}^\infty(1365k+181)\frac{S_k^{(2)}(1/18)}{16^k}=\frac{1377}{\sqrt2\,\pi}.$$

The source presents these as open conjectures motivated by Ramanujan-type series for $1/\pi$; no proof or resolution is supplied.