Does plan recompilation happens to plan guide? Does plan recompilation happens for plan guide due to the, for example, auto update stats?

P.S Do not take into account adding a hint OPTION (RECOMPILE) in the plan guide
As I understand it, a plan guide is not compiled like an execution plan. Therefore, it cannot be recompiled. That being said, a plan guide does result in an execution plan, either through its hints or by specifying a specific plan to use. The resulting or specified execution plan could be affected by auto update statistics.

If auto update statistics determines that a statistics update is necessary, query plans dependent on those statistics would be recompiled.