This is a Heyacrazy puzzle.
Rules of Heyacrazy:
Shade some cells of the grid.
Shaded cells cannot be orthogonally adjacent; unshaded cells must be orthogonally connected.
When the puzzle is solved, you must not be able to draw a line segment that passes through two borders, and does not pass through any shaded cells or grid vertices.
For an example puzzle and its solution, see this question.
Though I generally try to avoid requiring bifurcation or deep hypotheticals, that might be the easiest way to get the last few shaded cells. I would have liked to overclue it to make the ending clean, but when you solve the puzzle it should be clear why I couldn't do that.