While Bannister’s 4-minute mile record is used as an example of a psychological barrier, there’s also a reinterpretation of the meaning behind his record. Before his 1954 race, the record for the mile stood at just over 4 minutes (4:01.4) for 9 years. While speed records were set during WWII, they were all set by Swedish runners (Sweden being neutral in the war). The record today, which has stood since 1999, is 3:43.13. It's not a round number, so as a result gets less attention. Maybe that's why we don't think of it as a psychological barrier.

Reminds me of barriers in speedrunning. Technically all the times are arbitrary, but there's still prestige to be the first person to get under <nice number>. I don't think it really influences the speed of record breaking around it, except that time when there's literally a bounty raised.

so it's all a question of marketing

343 is 7 cubed, so just call it "cube barrier!" and it becomes a worthy challenge