The uniquely solvable sudoku code-golf decision-problem

The task

Given a standard 9x9 sudoku board, output a Truthy value if that sudoku admits one and only one solution. Output a Falsy value if the sudoku has a number of solutions other than one. This means 0 solutions and two or more solutions.

The input

The board can be given in any sensible format. Some come to mind, and I'll exemplify for a 4x4 sudoku.

• a 2D array with the state of the board, with any placeholder value for non-filled cells (orincluding the digit 0, or no value at all if your language supports it): [[1,2,#,4],[#,4,1,2],[2,1,4,#],[4,#,2,1]]
• a string of the digits row by row or column by column, so "12#4#412214#4#21" or "1#24241##14242#1"

The output

A Truthy value if the sudoku puzzle has a unique solution, Falsy otherwise.

Test cases

(To add)

Truthy

Falsy