Proposed norm inequalities for extending the radius estimate to $1<p\leq2$

Let $1<p<2$, set $r=2p/(2-p)$, and let $i_{p2}\colon\ell_p\to\ell_2$ be the formal identity map. Let $\pi_{r,2}(\cdot)$ denote the $(r,2)$-absolutely summing norm. For a linear map $B\colon\ell_2\to\ell_p$, let $\|B\|_{S_r}$ denote its $r$-Schatten norm,

$$\|B\|_{S_r}=\bigl(\operatorname{Tr}(B^*B)^{r/2}\bigr)^{1/r}.$$

For a positive semi-definite operator $A\colon\ell_{p'}\to\ell_p$, where $p'=p/(p-1)$, set $r=p/(2-p)$ when referring to the third inequality. Norm-inequality conjectures. The following statements are proposed as problems/conjectures:

(a) $\pi_{r,2}(i_{p2})\leq1$ for $p\in(1,2)$, with $r=2p/(2-p)$;

(b) $\|B\colon\ell_2\to\ell_p\|\geq\|B\|_{S_r}$ with $r=2p/(2-p)$;

(c) if $A$ is positive semi-definite, then $\|A\colon\ell_{p'}\to\ell_p\|\geq\|A\|_{S_r}$ with $r=p/(2-p)$ and $p'=p/(p-1)$.

These are presented as possible counterparts to the lemma used to obtain a sharp estimate on $\lambda'$ and are not resolved in the supplied text. They correspond to different proofs of the preceding norm lemma.