The ccc forcing limitation on clubsuit at the first uncountable cardinal

Let $V=L$. Let $\mathbb{P}$ be a ccc forcing notion and let $G\subseteq\mathbb{P}$ be generic. Write $\clubsuit_S$ for the clubsuit principle associated with a stationary set $S\subseteq\omega_1$. The ccc forcing limitation conjecture. If

$$2^\omega>\omega_1$$

in $V[G]$, then $\clubsuit_S$ fails in $V[G]$ for some stationary set $S\subseteq\omega_1$. The preceding discussion explains that this would show that the limitation on ccc forcing notions which increase the continuum and preserve clubsuit does not depend on large cardinals in the ground model; its status is not resolved in the supplied text.