Is the Projection Matrix estimated using least-squares pose specific?
Yes, that is correct.  If you move the camera, the projection matrix changes.

However, fortunately you don't have to start from scratch.  If you know the new position of the matrix relative to the old one, you can adjust the projection matrix.  Or, if you don't know how the camera has moved, you'll need to re-estimate the extrinsic parameters, but the intrinsic parameters will remain unchanged (the projection matrix is the composition of an intrinsic matrix and an extrinsic matrix, and only one depends on the camera's position).

See https://en.wikipedia.org/wiki/Camera_matrix and https://en.wikipedia.org/wiki/Camera_resectioning.