Can a creature kill itself by dashing during a chase?


DMG p.252

During the chase, a participant can freely use the Dash action a number of times equal to 3 + its Constitution modifier. Each additional Dash action it takes during the chase requires the creature to succeed on a DC 10 Constitution check at the end of its turn or gain one level of exhaustion.

Suppose a creature was in a chase and already at Level 4 of exhaustion. It chose to continue dashing and at the end of its turn failed its constitution check.

DMG p. 252

A participant drops out of the chase if its exhaustion reaches level 5, since its speed becomes 0.

Now suppose a creature that was [allowed to Dash twice on its turn] started its turn at exhaustion level 4 and chose to Dash twice. At the end of its turn, it would be required to make two Con checks. If it failed the first check and got to Exhaustion Level 5 would this immediately remove it from the chase and thus remove the need to check again, or would a second check still be required with the possibility of death if it failed?

Note that exhaustion from chases has a different recovery mechanic than other exhaustion, in that (DMG p. 252)

A creature can remove the levels of exhaustion it gained during the chase by finishing a short or long rest.

whereas exhaustion gained by other means requires a long rest, food, and drink. Thus there is some evidence that while chases are temporarily exhausting, the exhaustion gained from them is not as serious or as lasting. Might this include the chase-induced ‘death’ as well?