They believe they will, that the adversity will fuel them next season to go on another deep run.
The Lightning have reached the Stanley Cup Final once and lost in Game 7 of the Eastern Conference Final twice since 2015. Cooper and many of the core players from those teams are still around and signed long-term, including Hedman, Kucherov and captain Steven Stamkos.
Also signed for at least three more seasons are forwards Tyler Johnson, Ondrej Palat, Yanni Gourde, J.T. Miller and Alex Killorn, and defenseman Ryan McDonagh. Forwards Anthony Cirelli and Mathieu Joseph, and defensemen Erik Cernak and Mikhail Sergachev each have one year left on their entry-level contracts.
"We're built for years ahead," Hedman said. "We like the team that we have and the steps we've taken. The regular season wasn't a fluke. We're a good team."
They have the backing of their general manager, Julien BriseBois, who in the immediate aftermath of losing against the Blue Jackets said he would not overreact to the four-game sweep by making any drastic changes to the roster.
He has so far held true to that by re-signing defensemen Braydon Coburn to a two-year, $3.4 million contract and Jan Rutta to a one-year, $1.3 million contract.
The Lightning still have work to do to keep their core intact; center Brayden Point can become a restricted free agent on July 1. Vasilevskiy has one year remaining on his contract before he can become an RFA.
But BriseBois' words and actions have resonated with the players.
"When you hear that from your general manager it means a lot," Kucherov said. "It means they have confidence in us and one week doesn't change anything."
But one week did change the Lightning's perspective on what they need to do next season, provided they're good enough to get back into the Stanley Cup Playoffs.
"Yeah, we cannot get comfortable in the regular season," Vasilevskiy said.
Suggesting they can't get comfortable in the regular season again is another way of saying the Lightning did get comfortable this season. They were good enough to overcome it but wilted in the face of pressure in the Stanley Cup Playoffs.
That's the life they've been living for nine weeks. It's what they will live with all summer.
They have to go through it again: the offseason, training camp, exhibition schedule and 82 games to earn the chance to get another crack at it, to get rid of the would'ves, could'ves and should'ves that still plague them even when they could be honored for their accomplishments.
"It's a different story every year," Cooper said, "and the big thing for us is we want next year to be the story about us."