This myth persists even 13 years after federal law in the US explicitly forbid made discounts for cash or debit or ACH or whatever payment legal regardless of merchant agreements with card networks:

https://www.ftc.gov/business-guidance/resources/new-rules-el...

>Discounts to Customers

>A PCN cannot stop you from offering your customers a discount or another incentive for using a certain method of payment, as long as you offer it to all your customers and disclose the offer clearly and conspicuously. For example, you can offer your customers a discount or a coupon if they pay with cash or a debit card rather than a credit card. But the new rules do not address other PCN restrictions that may prevent you from offering discounts or similar incentives that vary based on the use of a card from a particular issuer or a particular PCN.

They may not be able to post different prices. But they'll often negotiate. Just ask!

I'll repeat: just ask!