Not the way these links explain it and I don't see Pence's name anywhere in the timeline. The links agree with the obstruction and dithering as testified by Capitol police Chief, Steven Sund, in the lead-up to J6, after Trump first requested 10,000 extra military personnel be on duty.
So somebody was lying, but it is not impulse.
https://www.politico.com/news/2021/12/06/jan-6-generals-lied-ex-dc-guard-official-523777
https://cha.house.gov/2024/9/transcripts-show-president-trump-s-directives-to-pentagon-leadership-to-keep-january-6-safe-were-deliberately-ignored