Convolution is a linear operation. So you can see it as (ignoring conv parameters): y = conv(x ,w).
Then dL/dx = dL/dy dy/dx = conv_transpose(dL/dy, w) and dL/dw = dL/dy dy/dw = conv(x, dL/dy). These new conv with modified parameters.
The way I check if the output is needed is by checking this file 10. That contains most of the derivative definitions.
There you can see that the various definition for convolution depend on the input/weight but never on result which is the output of the forward.