The endpoint absolute summability conjecture for the inclusion $X_p\subset L^p$

Let $1<p<2$. The space $X_p$ is a function space continuously embedded in $L^p$.

Endpoint absolute summability conjecture. The inclusion

$$X_p\subset L^p$$

is $(p,1)$-absolutely summing.

The paper proves the corresponding result for exponents $q>p$ and notes that boundedness on the relevant subspace would imply the endpoint statement. Whether the inclusion is $(p,1)$-absolutely summing remains open in the supplied text.