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In \cite{MV} we defined and proved the consistency of the principle ${\rmGM}^+(\omega_3,\omega_1)$ which implies that many consequences of strongforcing axioms hold simultaneously at $\omega_2$ and $\omega_3$. In this paperwe formulate a strengthening of ${\rm GM}^+(\omega_3,\omega_1)$ that we call${\rm SGM}^+(\omega_3,\omega_1)$. We also prove, modulo the consistency of twosupercompact cardinals, that ${\rm SGM}^+(\omega_3,\omega_1)$ is consistentwith ZFC. In addition to all the consequences of ${\rmGM}^+(\omega_3,\omega_1)$, the principle ${\rm SGM}^+(\omega_3,\omega_1)$,together with some mild cardinal arithmetic assumptions that hold in our model,implies that any forcing that adds a new subset of $\omega_2$ either adds areal or collapses some cardinal. This gives a partial answer to a question ofAbraham \cite{AvrahamPhD} and extends a previous result of Todor\v{c}evi\'{c}\cite{Todorcevic82} in this direction.

On Indestructible Strongly Guessing Models