Sudoku puzzle with exactly 3 solutions

While published sudoku puzzles typically have a unique solution, one can easily conceive of a sudoku puzzle with two solutions. However, is it possible to construct a sudoku puzzle with exactly 3 different solutions?

Inspired by https://puzzling.stackexchange.com/a/6.

Solutions Collecting From Web of "Sudoku puzzle with exactly 3 solutions"

You could use this as a model (this works for $4 \times 4$ but needs to be adapted for $9\times 9$):

$\begin{array}{cccc}
1 & 2 & & \\
3 & 4 & 1 & 2 \\
& 1 & & \\
& 3 & & 1 \end{array} $

With solutions

$\begin{array}{cccc}
1 & 2 & 3 & 4 \\
3 & 4 & 1 & 2 \\
2/4 & 1 & 4/2 & 3 \\
4/2 & 3 & 2/4 & 1 \end{array} $

and

$\begin{array}{cccc}
1 & 2 & 4 & 3 \\
3 & 4 & 1 & 2 \\
2 & 1 & 3 & 4 \\
4 & 3 & 2 & 1 \end{array} $