Update: Ran this again today. Modified the probabilities as follows: Kept 2⁄3 truth 1⁄3 lie, but had lookup-or-not be independently2⁄3 don’t lookup, 1⁄3 lookup. (So, we might look it up and then lie.) This differs from my earlier suggestion in that that was 1⁄2 don’t lookup, 1⁄2 lookup. That might work too. But I do feel like having lookup-or-not be independent of lying-or-not I do think was a good change.
Update: Ran this again today. Modified the probabilities as follows: Kept 2⁄3 truth 1⁄3 lie, but had lookup-or-not be independently 2⁄3 don’t lookup, 1⁄3 lookup. (So, we might look it up and then lie.) This differs from my earlier suggestion in that that was 1⁄2 don’t lookup, 1⁄2 lookup. That might work too. But I do feel like having lookup-or-not be independent of lying-or-not I do think was a good change.