We say that a subset $K$ of a Banach space $X$ is relatively $p$-compact ($1\leq p<\infty$) if there exists a $p$-summable sequence $(x_n)_{n=1}^{\infty}$ in $X$ such that $$ K\subseteq \left\{\sum_{n=1}^{\infty}\alpha_{n}x_{n}:(\alpha_{n})_{n}\in B_{l_{q}}\right\}\quad\quad(\frac{1}{p}+\frac{1}{q}=1).$$ A Bananch space $X$ is said to have the $p$-approximation property if $I_{X}$ can be approximated by finite rank operators uniformly on the relatively $p$-compact subsets of $X$. An operator $T:X \rightarrow Y$ is said to be quasi $p$-nuclear if there exists a $p$-summable sequence $(x^{*}_{n})_{n=1}^{\infty}$ in $X^{*}$ such that $$\|Tx\|\leq \left(\sum_{n=1}^{\infty}|\langle x^{*}_{n},x\rangle|^{p}\right)^{1/p} \quad\quad\hbox{ for all $x\in X$.} $$

We denote the Banach ideal of quasi $p$-nuclear operators by $\mathcal{Q}\mathcal{N}_{p}$ and the Banach ideal of $p$-summing operators by $\prod_{p}$. We say that a Banach space $X$ has the right approximation property with respect to $\mathcal{Q}\mathcal{N}_{p}$ (or $\prod_{p}$)if for every Banach space $Y$ and every operator $T\in \mathcal{Q}\mathcal{N}_{p}(X,Y)$(or $T\in \prod_{p}(X,Y)$), we have $T\in \overline{\{TS:S\in \mathcal{F}(X,X)\}}^{\tau}$, where $\tau$ is the topology of uniform convergence on compact subsets of $X$. My questions are:

If $X$ has the right approximation property with respect to $\mathcal{Q}\mathcal{N}_{p}$, does $X$ has the $p$-approximation property ?

If $X^{*}$ has the $p$-approximation property, does $X^{*}$ have the right approximation property with respect to $\prod_{p}$ ?