It depends on many factors, including AC or DC operation, voltage, current, solenoid inductance, relay contact construction and relay contact material.
First: your show your solenoids having only 1 uH inductance. I don't believe that is anywhere near close to their actual value - several hundred milli-Henrys on through several Henrys would be more like it.
My company manufactures a lot of circuit boards used to build HVAC equipment where we have relays driving water circulating pumps that run at AC Mains voltage (both 120 Vac & 230 Vac). Those relays are NAIS / Aromat JS1 series and we have shipped boards containing several hundred thousand relays over the past 20 years or so. There is NO transient suppression on the contacts. I am not aware of any relay contact failures that weren't caused by installer wiring errors.
A different class of board uses either 1 or 3 American Zettler AZ2150 & AZ2150A relays directly controlling 3/4 HP induction motors (HVAC Blowers). Same deal there: no relay failures not caused by installer wiring errors.
In general, I worry about requiring arc suppression on relays contacts driving DC inductive loads. It seems to be much less of an issue when driving inductive loads from an AC source.
That said: I examine each specific application carefully at the design stage and determine if relay contact transient suppression is warranted.