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The 29-year-old looks set to miss his first game of the season on Saturday, leaving the Rovers boss with a hole to fill.
Jayden Fevrier took his spot as a subtitute at The Hawthorns and will be pushing to start. Augustus Kargbo and Mathias Jorgensen are also options if Mowbray opts to go like-for-like.
The alternative is a reversion to 3-4-3, with Morishita's injury easing the headache as to which forward would have to miss out.
Given Wrexham's aerial prowess and physical forwards, it wouldn't be a shock to see Rovers go with three centre-backs, matching Wrexham's shape too.
That would mean a return for Hayden Carter alongside Tom Atcheson and Sean McLoughlin in the back three. The wing-backs would stay the same, as would the central midfield pairing.
It would create a slight imbalance in attack with Todd Cantwell having to operate wide right out of possession, though he could drift centrally to create when they have the ball.
Predicted Rovers XI: Toth; Alebiosu, Atcheson, Carter, McLoughlin, Pickering; Morsy, Montgomery; Cantwell, Afolayan; Gudjohnsen.