A Falcoz digit-fancy

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In addition to my own "fanciful extension" of Éric Angelini's Two identical digits effort, Jean-Marc Falcoz suggested his own variation (at the end of the blog entry): "Lexicographically earliest sequence of distinct positive terms such that [the product of adjacent terms] contains exactly 1 digit 1 (if 1 is present), 2 digits 2 (if 2 is present), 3 digits 3 (if 3 is present), ... 9 digits 9 (if 9 is present)." He presented 113 terms of the sequence but I was hungry for more.

The above plot just exceeds 1000 terms. Surprisingly, term #318 is 17 and term #319 is 13, local minima. Term #455 is 1011211671, a thus-far maximum. Possible products are given by A108571. Our indexed products are such that product #2 is term #2 multiplied by term #1 (product #1 is 1 by fiat). In the current list there is only one duplicate: product #172 = product #622 = 2423433144. If typed by their constituent digits, regardless of digit order, the number of possible types is given by A125573. Our current list realizes just 71, the number of which (sorted by product digit-length) are: 1, 1, 1, 2, 0, 0, 2, 3, 2, 4, 5, 7, 9, 11, 9, 10, 4.