I assume this rose would sucker if grown on its own roots?

Seeing the green button in the flower, I understand that it can't set seeds... But does it have fertile pollen?
This would seem such an interesting rose to breed from! But i have heard of it's sterility.

Just an observation from this summer: my 2nd year plant was over 9 ft tall, and I pruned it rather hard after flowering back to approx. 3 ft. It did not like that at all. It dropped almost all its remaining leaves, and perhaps half the canes died. I began watering it profusely and it did slowly respond, but now at the end of the growing season, it has very little new growth, and I’m doubting survival over the winter.

Update August 2023: Madame did survive the 2022 - 23 winter, but emerged late in the spring, and showed no interest in blooming. However, by late June she was growing vigorously, and throwing up new basal canes. So my new “pruning” regimen is pegging the longest canes, and making weekly snips wherever needed to keep her in bounds. Oh, the things we do for love of a rose!