Prefix-exchange conjecture for increasing and decreasing patterns

Let $\sigma$ and $\tau$ be patterns, and write $\sigma\sim_{\mathcal{I}}\tau$ when they are equinumerous for avoidance by involutions, meaning that $\mathcal{I}_{n}(\sigma)=\mathcal{I}_{n}(\tau)$ for every $n$. A prefix is an initial consecutive pattern such as $12\ldots k$ or $k\ldots21$. Prefix-exchange conjecture. For every $k$, the prefixes $12\ldots k$ and $k\ldots21$ may be exchanged; equivalently,

$$12\ldots k\sim_{\mathcal{I}}k\ldots21.$$

The conjecture extends the known results for short increasing and decreasing prefixes and would explain further pattern equivalences for involutions. Its general case is left open in the source.