The smoothability conjecture for finitely generated PL groups

Let $G\le \operatorname{\mathsf{PL}}([0,1])$ be a finitely generated group of piecewise linear homeomorphisms of $[0,1]$. Its standard action is assumed to be conjugate to a $C^\infty$ action. Smoothability conjecture. Then $G$ embeds in Thompson's group $F$. This conjecture is motivated by the contrast between groups whose standard actions can be smoothed and higher-rank slope groups, which admit no faithful $C^r$ action for any $r>1$.