Conjectured formulas for three-rowed tableau major-index distributions

Let $f_{\lambda,i}(q)=\sum_{\tau\in SYT(\lambda):\,\operatorname{des}(\tau)=i}q^{\operatorname{maj}(\tau)}$, where $SYT(\lambda)$ is the set of standard Young tableaux of shape $\lambda$. For the indicated partition shapes, the following formulas are conjectured:

$$f_{(n,n,3),3}=q^{n+8}\left[{n+2\atop1}\right]_q\left[{n\atop3}\right]_q\left[{2\atop1}\right]_q,$$ $$f_{(n,4,4),3}=q^{14}\left[{n-2\atop2}\right]_q\left[{n\atop1}\right]_q\left[{6\atop1}\right]_q-q^{17}\frac{(1-q^4)(1-q^{n-3})^2(1-q^{n-2})}{(1-q)^2(1-q^2)^2}.$$

Conjectured three-rowed formulas. The two displayed identities hold.

These identities are proposed as terminal cases for a recurrence intended to produce formulas for broader families of three-rowed tableaux. The supplied text gives no proof or resolution status.