Is there any printed or handwritten precursor to this 门 (door, gate) glyph?
There is such handwriting in calligraphy works of Tang Yin (唐寅, also well known as 唐伯虎, Tang Bohu, 1470-1524, Ming Dynasty). 

I found some pictures of his writing, 《落花诗册》.  I marked the related characters with a mark to the right. The genuine writing is now in Suzhou Museum, Jiangsu Province.