I think she's dead, but mainly so because otherwise my headcanon of Jigoro's Katana wouldnt work
I like to think that the katana he wears around his waist was from his father's old sword collection, and his wife's personal favourite.
Around the time of her death, the Hakamichi family was close to the poverty line. Jirogo had an idea for a business and his wife was all for it, but they didn't have the funds to get started. They were going to run the business together, hand in hand, sharing the same dream... Unfortunately, she came down with a serious case of a disability known as "being dead".
In a last bid to fulfil their dream, Jigoro made the tough decision to sell their family's antique sword collection. He sold all of them but one, the one his wife loved.
They sold for quite a bit, but only just enough to start a small workshop in his cramped house. Over the next few years he put the business above all else, all in her memory, not realising how much he was neglecting his own children. He has worked hard for everything he has, every last penny... Even though they don't use pennies in Japan.
His overmasculinity comes from the idea that his wife liked to see his 'manly side' from time to time, and would playfully make a fuss when he made a campfire or intimidated someone who tried to chat her up. In her death, almost as if trying to make her proud, he refused to be anything less than pure alpha. He hasn't actually realised how much of a douche he comes across as; as far as he's concerned he's just acting the way she liked him to... not to say she would necessarily agree if she could see him now