With a bit of experimentation with the Amp Envelopes (Amp EG), you might be able to get what you need.
In theory, you'll need to reduce the Amp EG's Sustain Level stage to 0 to prevent endless sustain, modify the AMP EG Break stage to give some amount of sustain when the keys are held, and use the Damper pedal as an AMS Source in the EG Time Modulation section to modify both the Break and the Release stages (I'll give an example below). It's important to note that the modifications won't happen to keys already pressed down if you step on the damper afterward. You'll need to step on the damper pedal first and then play the keys for the changes to work.
As an example (you'll want to experiment with the settings to suit the effect you want), call up a string sound in Program mode.
Navigate to the AMP1 EG tab (see pg 86 in the Parameter Guide for ref).
In the Envelope section, in the row for Level, adjust the Sustain to 0.
Next, in the row for Time adjust the Slope to a high value, like 90 or so.
Play keys and keep them held down, and ensure you get some amount of sustain.
If you're happy with that, move on to the Time Modulation section.
Hopefully, there is a unused AMS here... typically AMS3. If not, you'll need to re-purpose one that is already used for use with the Damper pedal. Let's use AMS3.
Tap the arrow next to AMS3 and select Damper from the list.
Set the Slope value to something close to -45.
Set the Release value to something close to +45.
Ensure the other settings here are set to a value of 0.
Now, step on the damper pedal. Play the keys with a staccato action. This should give you an ending sustain. Adjust and experiment with the Slope and Release values to taste.
Important: If your string sound uses two oscillators (very likely), you may need to duplicate these settings on the AMP2 EG tab.
Here's an image of the settings you'll need to adjust -- this just shows the location of the settings, not the values described above:
