An estimate of a series with exponentials

Question

Provide an example of a sequence $(p_n)_{n=1}^\infty\subset(0,1)$ such that $$ \sum_{n=1}^\infty\left(\frac23\right)^n\frac{p_n}ne^{-s3^{-n}(1-p_n)/2}=o(s^{-2}),\qquad s\to\infty. $$ (From some geometric argument it's clear to me that such a sequence should exist, but I would need a concrete example.)