In Sheviis 8:9:
עוֹר שֶׁסָּכוֹ בְּשֶׁמֶן שֶׁל שְׁבִיעִית, רַבִּי אֱלִיעֶזֶר אוֹמֵר, יִדָּלֵק. וַחֲכָמִים אוֹמְרִים, יֹאכַל כְּנֶגְדּוֹ. אָמְרוּ לִפְנֵי רַבִּי עֲקִיבָא, אוֹמֵר הָיָה רַבִּי אֱלִיעֶזֶר, עוֹר שֶׁסָּכוֹ בְשֶׁמֶן שֶׁל שְׁבִיעִית, יִדָּלֵק. אָמַר לָהֶם, שְׁתֹקוּ, לֹא אוֹמַר לָכֶם מַה שֶּׁרַבִּי אֱלִיעֶזֶר אוֹמֵר בּוֹ:
Animal hide which is anointed in Shemitah oil, Rabbi Eliezer says: it must be lit on fire. And the Sages say: he must eat a corresponding amount [of other food as if it were Sabbatical produce]. They said in front of Rabbi Akiva: Rabbi Eliezer would say, "Animal hide which is anointed in oil [produced during] the Sabbatical year must be burned." He responded to them: “Quiet! I will not tell you what Rabbi Eliezer says about this.”
Says the Yachin there:
דמיקל בה טפי ולא רצה לגלות
[R’ Akiva] was much more lenient and did not want to tell them.
We have a principle that we go according to the majority opinion (see, for instance, Bava Metzi’a 59b). So why didn’t R’ Akiva just tell them that he doesn’t hold like R’ Eliezer because the majority of Rabbis disagree with him?