Does the forcing PR* that adds a bounding-predictor keep e small?

Ascertain whether the forcing PR*, which generically adds a bounding-predictor and has ultrafilter-limits, also preserves small values of the ordinary evasion number e under iteration.

Background

The author proposes a natural modification PR* of the prediction forcing that adds a bounding-predictor. It is shown that PR* has ultrafilter-limits, hence it keeps the bounding number b small.

What remains unresolved is whether PR* also keeps the ordinary evasion number e small, a key requirement to achieve certain separations involving e and e*.