2 Questions on Rep-tiles

A rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape.

Question 1: Are there any convex pentagons that are also rep-tiles?

Remarks: 15 convex pentagonal tiles of the plane are known and none of them appears to be a rep-tile. Assuming this observation is right, one can invoke a proof given in 2017 by Michel Rao - that these 15 are the only convex pentagonal tiles possible - to answer our question in the negative. However, I don't know if Rao's proof has been validated and if there is a simpler (elementary) proof that there are say no convex pentagonal rep-tiles. Basically, one is asking for a simpler proof for a weaker claim.

Question 2: Let us define a *multi-way rep-tile* as a polygon *P* with the property: if *P1* and *P2* are larger scaled up copies of *P* and *P1* can be tiled with *m* units of *P* and *P2* can be tiled with *n* units of *P* with *n*> *m*, then, a layout of *n* *P*-units can form *P_2* without *m* of the units in the layout together forming a *P_1*. As shown on this page: https://en.wikipedia.org/wiki/Rep-tile, there are isosceles trapeziums with angles 60 and 120 degrees with multi-way property (with *m*= 4 and *n* = 9). Are there other convex polygons with this multi-way rep-tile property?

Note: A square is obviously a rep-tile but is not multi-way.