Yes, the pelvic joint at the back is loose. The whole pelvis front and back can be affected and can strain your lower spine. Your hips maybe struggling too dependent on how deep your socket is. The hip pain is like a bone ache that penetrates your thighs.
But Ive joined up to tell you that the only thing that has massively improved my SPD was taking vitamin d. Here's what I've tried in the years I've had it...During pregnancy used a support belt, limited movement and a group physio session that gave us tips on limiting movement. As you know yourself, this doesn't tackle the issue. You're just trying not to aggravate it. Post birth it carried on so I went to physio by myself and was prescribed exercises. No improvement. Still walking like I needed a hip replacement. Getting desperate after a year of this I paid for a private ostheopath who massaged the tightened muscles but the underlying pelvic dysfunction obviously wasn't going to disappear (though I didn't know thus at the time). Interestingly, they said the physio exercises were all wrong and actually tightening an area that needed loosening. That made sense as the physio hadn't worked. Went to pilates but no real improvement although that was good for understanding what my body could now do (or not do).
Finally, accepted it would always be crap. Until last year, when I went and had blood tests as I was by now quite fatigued and my vit d level was low. Borderline low, so not massively. So I was advised to take vitamin d to raise it (not in relation to SPD). And within weeks I could feel my pelvis come back together. I can only surmise that what caused the loosening of my ligaments was caused by low vit d levels.
I would ask your gp if they can check your level and prescribe vit d if it is at the lower end. My gp, midwife, physio, physio team at hospital and osteopath never mentioned it as a possible cause. Which is a shame because when they finally xrayed my hips last year just before I started taking the vitamins, my hip sockets had narrowed as a result of the wear and tear of my unstable pelvis. That said, I am hugely improved. As an aside, I also suffered increasingly from IBS and that has also calmed right down since taking vit d.
Obviously, you should only take it if it's prescribed by a gp as it's actually a hormone I believe.
I hope my experience helps...meantime osteopath massage and Pilate's, along with limited movements will help you manage it but won't fix it.