I don't think so, the average tx is still above 5,000
Let the free market be.
More people will be moving Sats money on LN, On-Chained will be used when they become wealthy enough.
 

 This. Can't expect people to pay such fees for small transfers. 

 Small transfers are done on LN, the bigger transfers are done On-Chain. 
No body mind paying $10 to move $10K, but they need to move $5 they can simply send it over LN for 1 Sat