The monotonicity conjecture for ternary cubic multinomial sums

For $n\in\mathbb N$, define

$$S_3^{(3)}(n)=\sum_{k_1+k_2+k_3=n}\binom{n}{k_1,k_2,k_3}^{3}.$$

Monotonicity conjecture for $S_3^{(3)}$. The sequence

$$\left(\frac{S_3^{(3)}(n+1)}{S_3^{(3)}(n)}\right)_{n\geq0}$$

is strictly increasing and converges to $27$, while the sequence

$$\left(\frac{\sqrt[n+1]{S_3^{(3)}(n+1)}}{\sqrt[n]{S_3^{(3)}(n)}}\right)_{n\geq1}$$

is strictly decreasing and converges to $1$. The conjecture is presented as being based on computation.