Step 1: Identify what fringe width means. Fringe width is the distance between two neighbouring bright (or dark) fringes, given by \(\beta = \lambda D/d\).
Step 2: Analyse the role of the sheet through path difference. Without the sheet the path difference at a point is \(\Delta = d\sin\theta\). With a sheet of thickness \(t\) and refractive index \(\mu\) over one slit, it becomes \(\Delta' = d\sin\theta - (\mu-1)t\), i.e. a fixed offset is subtracted.
Step 3: Find the new fringe positions. Bright fringes still occur at equal steps of \(\Delta' = n\lambda\); solving, consecutive fringes are still separated by the same \(\lambda D/d\), only the whole set is displaced by \((\mu-1)tD/d\).
Step 4: Because the separation between successive maxima is untouched, the fringe width does not change, matching option (iii). Only the position of the pattern shifts.
\[\boxed{\text{Fringe width remains } \lambda D/d}\]