Equivalence from multi-tape to single-tape implies limited write space?
Sipser's book handles this case:


  If at any point $S$ moves one of the virtual heads to the right onto a $\#$, this action signifies that $M$ has moved the corresponding head onto the previously unread blank portion of that tape. So $S$ writes a blank symbol on this tape cell and shifts the tape contents, from this cell until the rightmost $\#$, one unit to the right. Then it continues the simulation as before.