For those who pronounce a דגש חזק as a double letter (e.g. הַמֶּלֶךְ as הַמְמֶלֶךְ but with a שוא נח on the first מ''ם), how should a דגש חזק at the beginning of a word be pronounced (e.g. הוֹשִׁיעָה נָּא; לְמַעְלָה רֹּאשׁ; etc.)?
According to Yeivin, the value of the conjunctive dagesh is uncertain. When it follows a short vowel, it could naturally behave as a strong dagesh. However, when conjunctive dagesh follows a consonant (e.g. Josh. 5:14 according to Ben Naftali; מִשְׁכָּנֹ֥ות לֹּא־לֹֽו in Hab. 1:6 or עַל־רִ֥יב לֹּֽא־לֹֽו in Prov. 26:1; Gen. 24:36 וַיִּתֶּן־לֹּ֖ו) or a long vowel it is harder to argue that it has a geminatory ability. Based on this, he believes that the conjunctive dagesh does not close the previous syllable (Introduction to the Tiberian Masorah, pp. 295-296).
Some texts with Babylonian vocalization write dehiq/ate merahiq as a dot between two words, and not as a regular dagesh (ג) (Kahle Masoreten des Ostens, p. 13). Moreover, the conjunctive dagesh is sometimes marked by Ben Naftali where Ben Asher marks a paseq (Yeivin p. 303).
Still, the Karaite Arabaic transcriptions, which are known to correspond to the Tiberian pronunciation well, show a shadda (gemination marker) on some instances of dehiq/ate merahiq. Moreover, the Secunda of the Hexapla has μεββεσε for מַה־בֶּ֥צַע (Ps. 30:10). Generally, pronunciation traditions that preserve consonant gemination geminate the conjunctive dagesh.