The Poincar\'e Inequality does not improve with blow-up

For each $\beta >1$ we construct a family $F_{\beta }$ of metric measure spaces which is closed under the operation of taking weak-tangents (i.e.~blow-ups), and such that each element of $F_{\beta }$ admits a $(1,P)$-Poincar'e inequality if and only if $P>\beta$.