Number Cross 5 | PuzzledQuant

This 6×6 grid is split into five regions, named A to E below. The letters are only names. Two cells, marked *, are highlighted.

        c0  c1  c2  c3  c4  c5     clue for the row
  r0    A   A   A   A*  A   A      multiple of 17
  r1    A   B   B   B   A   A*     product of digits is 45
  r2    A   B   C   C   C   A      product of digits is 12
  r3    D   D   C   C   E   E      multiple of 7
  r4    D   D   D   E   E   E      product of digits is 20
  r5    D   D   D   E   E   E      multiple of 19

Rules:

Put a digit 1-9 in every cell. All cells of a region hold the same digit, and orthogonally adjacent cells in different regions hold different digits. So any two regions that share an edge get different digits; in this grid that is every pair except B and E (for example, B and D touch at r2c1/r3c1).
Then place some tiles by blacking out cells. No two tiles may share an edge. A tile displaces the digit of its cell: the tile's orthogonal neighbours, taken together, are incremented by exactly that amount. You choose the split, and a neighbour may receive 0. A tile on the edge of the grid simply has fewer neighbours. A cell next to two or more tiles may receive increments from each of them, and its final value is its digit plus all of these. No cell may go above 9. After the increments, adjacent cells of different regions may end up equal.
A highlighted cell may not hold a tile and may not be incremented. A tile may still sit next to a highlighted cell; it just sends that cell 0.
In each row, every maximal run of un-tiled cells is read left to right as a number. Every such number must satisfy that row's clue and be at least two digits long. No number may appear twice anywhere in the grid.

There is exactly one completed grid.

Task. Find the sum of all the numbers formed in the completed grid.

Answer format: a single integer (for example 12345).

This 6×6 grid is split into five regions, named A to E below. The letters are only names. Two cells, marked *, are highlighted.

        c0  c1  c2  c3  c4  c5     clue for the row
  r0    A   A   A   A*  A   A      multiple of 17
  r1    A   B   B   B   A   A*     product of digits is 45
  r2    A   B   C   C   C   A      product of digits is 12
  r3    D   D   C   C   E   E      multiple of 7
  r4    D   D   D   E   E   E      product of digits is 20
  r5    D   D   D   E   E   E      multiple of 19

Rules:

Put a digit 1-9 in every cell. All cells of a region hold the same digit, and orthogonally adjacent cells in different regions hold different digits. So any two regions that share an edge get different digits; in this grid that is every pair except B and E (for example, B and D touch at r2c1/r3c1).
Then place some tiles by blacking out cells. No two tiles may share an edge. A tile displaces the digit of its cell: the tile's orthogonal neighbours, taken together, are incremented by exactly that amount. You choose the split, and a neighbour may receive 0. A tile on the edge of the grid simply has fewer neighbours. A cell next to two or more tiles may receive increments from each of them, and its final value is its digit plus all of these. No cell may go above 9. After the increments, adjacent cells of different regions may end up equal.
A highlighted cell may not hold a tile and may not be incremented. A tile may still sit next to a highlighted cell; it just sends that cell 0.
In each row, every maximal run of un-tiled cells is read left to right as a number. Every such number must satisfy that row's clue and be at least two digits long. No number may appear twice anywhere in the grid.

There is exactly one completed grid.

Task. Find the sum of all the numbers formed in the completed grid.

Answer format: a single integer (for example 12345).