K (ngn/k), 37 bytes
{+/x{{(a~&\a)|a~|\a:10\x}(1+)/x+1}\0}
{ } is a function with argument x
x{ }\0 applies the {} on 0 x times, preserving the intermediate results
(1+) is the successor function
{ }(1+)/x+1 applies the successor function starting from x+1 until the {} returns true
10\x are the decimal digits of x
a: assign to a
|\ is the max-scan (partial maxima) of a
&\ analogously, is the min-scan
a~|\a does a match its max-scan?
| or
a~&\a its min-scan?
+/ sum