Abstract In [15] we defined and proved the consistency of the principle upper G upper M Superscript plus Baseline left parenthesis omega 3 comma omega 1 right parenthesis $\mathrm {GM}^+(\omega _3,\omega _1)$ G M + ( ω 3 , ω 1 ) which implies that many consequences of strong forcing axioms hold simultaneously at omega 2 $\omega _2$ ω 2 and omega 3 $\omega _3$ ω 3 . In this paper we formulate a strengthening of upper G upper M Superscript plus Baseline left parenthesis omega 3 comma omega 1 right parenthesis $\mathrm {GM}^+(\omega _3,\omega _1)$ G M + ( ω 3 , ω 1 ) that we call upper S upper G upper M Superscript plus Baseline left parenthesis omega 3 comma omega 1 right parenthesis $\mathrm {SGM}^+(\omega _3,\omega _1)$ S G M + ( ω 3 , ω 1 ) . We also prove, modulo the consistency of two supercompact cardinals, that upper S upper G upper M Superscript plus Baseline left parenthesis omega 3 comma omega 1 right parenthesis $\mathrm {SGM}^+(\omega _3,\omega _1)$

Abstract Assuming $\mathrm{PFA}$ , we shall use internally club $\omega _1$ -guessing models as side conditions to show that for every tree T of height $\omega _2$ without cofinal branches, there is a proper and $\aleph _2$ -preserving forcing notion with finite conditions which specialises T. Moreover, the forcing has the $\omega _1$ -approximation property.