I don't have any scientific proof but my observations.
Firstable to clear the points Delilah talks about a plant that desperately needs new roots.

All the above that Ray said I think its true about healthy plants but if plant has the spike and no or little roots to support it its in plant's best interest to cut spike off even if flowers are still nice.

Think about it that all water/supplements roots can get have to be shared between leaves and the spike. So by cutting spike you let those recourses to go to leaves so they can photosynthesis all necessary stuff for plant living. The more stuff is in leaves the more those leaves can support the whole plant until it will grow new roots.

So my observation is that (on healthy plants) existing roots are still growing while plants are in spike but don't produce new root growth. As soon as most of flowers or last flowers are gone you can see new leaf forming and new roots showing up.
The people who using keiki paste talk about research made about hormones produced by the plant during flowering stage and vegie (leave growing) stage. So yes the plant sends info if its time to leaf/root growth.

By cutting spike you will decrease flowering hormones and plant will think its time for veggie stage.

Also Delilah, you don't give us enough info, in what exactly condition are existing roots?

if there is only one healthy root for big leave mass and the spike than spike will suck all nutrient from leaves to support flowers, but at the end you might loose whole plant. So you have to make decision what you want to save plant or flowers.
You can cut spike and keep it an a vase for weeks and save whole plant. I was there, done that, now phal puts new roots